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We study the problem of predicting the future, though only in theprobabilistic sense of estimating a future state of a time-varying probabilitydistribution. This is not only an interesting academic problem, but solvingthis extrapolation problem also has many practical application, e.g. fortraining classifiers that have to operate under time-varying conditions. Ourmain contribution is a method for predicting the next step of the time-varyingdistribution from a given sequence of sample sets from earlier time steps. Forthis we rely on two recent machine learning techniques: embedding probabilitydistributions into a reproducing kernel Hilbert space, and learning operatorsby vector-valued regression. We illustrate the working principles and thepractical usefulness of our method by experiments on synthetic and real data.We also highlight an exemplary application: training a classifier in a domainadaptation setting without having access to examples from the test timedistribution at training time.

Predicting the Future Behavior of a Time-Varying Probability Distribution